Question

Wood Specialties Company produces wall shelves, bookends, and shadow boxes. It is necessary to plan the production schedule for next week. The wall shelves, bookends, and shadow boxes are made of oak, of which the company has 600 board feet. A wall shelf requires 4 board feet, bookends require 2 board feet, and a shadow box requires 3 board feet. The company has a power saw for cutting the oak boards into the appropriate pieces; a wall shelf requires 30 minutes, bookends require 15 minutes, and a shadow box requires 15 minutes. The power saw is expected to be available for 36 hours next week. After cutting, the pieces of work in process are hand finished in the finishing department, which consists of 4 skilled and experienced craftsmen, each of whom can complete any of the products. A wall shelf requires 60 minutes of finishing, bookends require 30 minutes, and a shadow box requires 90 minutes. The finishing department is expected to operate for 40 hours next week. Wall shelves sell for $29.95 and have a unit variable cost of $17.95; bookends sell for $11.95 and have a unit variable cost of $4.95; a shadow box sells for $16.95 and has a unit variable cost of $8.95.

Formulate this as an LP problem. Clearly describe your decision variables, objective function and constraints.

Answer 1

Let,

Number of wall shelves to produce = x1
Number of bookends to produce = x2
Number of shadow boxes to produce = x3

Objective is to maximize profit = max (29.95-17.95)x1+(11.95-4.95)x2+(16.95-8.95)x3 = max 12x1+7x2+8x3

Subject to,

4x1+2x2+3x3 <= 600 (Oak)
(30/60)x1+(15/60)x2+(15/60)x3 <= 36 (Power Saw)

or, 0.5x1+0.25x2+0.25x3 <= 36
(60/60)x1+(30/60)x2+(90/60)x3 <= 40 (Finishing)

or, x1+0.5x2+1.5x3 <= 40

x1,x2,x3 >= 0 (non-negativity constraint)

Solving in solver we get,

Number of wall shelves to produce = x1 =0
Number of bookends to produce = x2 = 80
Number of shadow boxes to produce = x3 = 0

maximized profit = 560

Solver screenshot

Solver window

Solver formula